Multidimensional diffeomorphisms with stable periodic points

Abstract

Diffeomorphisms of a multidimensional space into itself with a hyperbolic fixed point are considered, it is assumed that at the intersection of the stable and unstable manifolds there are points that are different from the hyperbolic, such points are called homoclinic. The homoclinic points are divided into transversal and non-transversal, depending on the behavior of stable and unstable manifolds. From the articles of S. Newhouse, L. P. Shil’nikov, B. F. Ivanov and other authors, it follows that with a certain method of tangency of the stable manifold with unstable one at the homoclinic point, a neighborhood of a non-transversal homoclinic point contains an infinite number of stable periodic points, but at least one of the characteristic exponents at these points tends to zero with increasing period. The proposed work is a continuation of the work of the author. In previously published papers, restrictions were imposed on the eigenvalues of the Jacobi matrix of the original diffeomorphism at a hyperbolic point. More precisely, in these papers it was assumed that either all eigenvalues are real and the matrix is diagonal, or the Jacobi matrix has only one real eigenvalue modulo less than one, and all other eigenvalues are different complex integers modulo greater than unity. Under these conditions, conditions are obtained for the presence in an arbitrary neighborhood of a non-transversal homoclinic point of an infinite set of stable periodic points with characteristic exponents separated from zero. In this paper, it is assumed that the Jacobi matrix of a diffeomorphism has an arbitrary set of eigenvalues at a hyperbolic point. In this case, the conditions of existence in the neighborhood of the non-transversal homoclinic point of an infinite set of stable periodic points, whose characteristic exponents are separated from zero, are obtained. The conditions are imposed, first of all, on the method of tangency of the stable manifold with unstable one; however, in the proof of the theorem, the properties of the eigenvalues of the Jacobi matrix at a hyperbolic point are essentially used.

Keywords:

multidimensional diffeomorphism, non-transversal homoclinic point, stability, characteristic exponents separated from zero